For your 8 and 5 years olds, you will need to make in-year applications for state schools - so you need to talk to the local education authority (LEA -in this case Surrey) as soon as you are resident in the county and have a provable address.
Where are you now? If you are forces, you can apply from outside the county in advance as soon as you have an address here, but if not, then you can apply from overseas/elsewhere up to 4 weeks ahead of coming here but your local address will not be taken into consideration until you are actually resident in county.
For your 3 year old, depending on when they turn 4, (if they do so before Sept 1, 2017) you would need to make a late application for a reception place.
Some schools manage their own in-year applications process, the surrey website shows you which; for those you apply direct to the school, others to the LEA.
www.surreycc.gov.uk/schools-and-learning/schools/school-admissions/apply-for-a-school-place/in-year-admissions/how-to-apply-for-an-in-year-school-place
Essentially, you'll get given places where there are spaces free in the relevant years. There is no guarantee they will be in the same school though. If they are placed separately, you can put them on waiting lists, or appeal for a place in another school - though appeals are hard to win, especially in reception/yr1/yr 2 if there is already 30 children per class as legislation prevents normal admissions numbers going over this.
For private schools you would need to talk to the schools individually, but a primary school place would probably cost 8-10k per child per year.